Solve: [tex]12^{x^2+5 x-4}=12^{2 x+6}[/tex]
Answer (2)
Equate the exponents: x 2 + 5 x − 4 = 2 x + 6 .
Rearrange the equation: x 2 + 3 x − 10 = 0 .
Factor the quadratic equation: ( x + 5 ) ( x − 2 ) = 0 .
Solve for x: x = 2 , x = − 5 . The solution is x = 2 , x = − 5 .
Explanation
Equating the exponents We are given the equation 1 2 x 2 + 5 x − 4 = 1 2 2 x + 6 . Since the bases are equal, we can equate the exponents. This gives us the equation x 2 + 5 x − 4 = 2 x + 6 .
Rearranging the equation Now, we rearrange the equation to form a quadratic equation. Subtracting 2 x and 6 from both sides, we get x 2 + 5 x − 4 − 2 x − 6 = 0 , which simplifies to x 2 + 3 x − 10 = 0 .
Factoring the quadratic equation Next, we factor the quadratic equation. We are looking for two numbers that multiply to − 10 and add to 3 . These numbers are 5 and − 2 . So, we can factor the quadratic equation as ( x + 5 ) ( x − 2 ) = 0 .
Solving for x Finally, we solve for x . Setting each factor equal to zero, we have x + 5 = 0 or x − 2 = 0 . Solving these equations, we find x = − 5 or x = 2 . Thus, the solutions are x = 2 and x = − 5 .
Final Answer Therefore, the solutions to the equation 1 2 x 2 + 5 x − 4 = 1 2 2 x + 6 are x = 2 and x = − 5 .
Examples
In real-world scenarios, exponential equations like the one we solved can model various phenomena, such as population growth, radioactive decay, and compound interest. For instance, if we have an investment that grows exponentially, we can use such equations to determine when the investment will reach a certain value. By setting up the equation and solving for the variable, we can predict future outcomes and make informed decisions.
We equated the exponents from the given equation, rearranged it to form a quadratic equation, then factored and solved it. The solutions are x = 2 and x = − 5 . Therefore, the answers to the equation are these two values of x .
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